operated by los alamos national security, llc for nnsa

Operated by Los Alamos National Security, LLC for NNSA

U N C L A S S I F I E DLANS Company Sensitive — unauthorized release or dissemination prohibited Slide 1

Relativistic U(3) and pseudo-U(3) symmetry of the Dirac Hamiltonian

Joseph N. GinocchioTheoretical Division, Los Alamos National Laboratory

JNG, PRC 69, 034318 (2004) JNG, Phys. Rep 414, 165 (2005)

JNG, PRL 95, 252501 (2005)

NEW QUESTS IN NUCLEAR STRUCTURE

VIETRI SUL MARE, MAY 21-25, 2010


U N C L A S S I F I E DLANS Company Sensitive — unauthorized release or dissemination prohibited

The Dirac Hamiltonian



Pseudospin Symmetry occurs in the spectrum of nuclei PRL 78, 436 (1997)

Vs – Vv = Cs Spin Symmetry Vs + Vv = Cps Pseudo-Spin Symmetry

The Dirac Hamiltonian has an invariant SU(2) symmetry in two limits:



VS (rr ) =

%M2 i=1

3

∑ ω 2S,ixi

2

VV (rr ) =

%M2 i=1

3

∑ ω 2V ,i xi

2

Relativistic Harmonic Oscillator

In the symmetry limits the eigenfunctions and eigenenergies can be solved analytically. The eigenfunctions are similar to the non relativistic limit. That is, the upper and lower components can be written in terms of

Gaussians and Laguerre polynomials. This true for spherical and non-spherical harmonic oscillator. JNG, PRC 69, 034318 (2004)

VS (rr ) =

%M2 i=1

3

∑ ω 2S,ixi

2

VV (rr ) =

%M2 i=1

3

∑ ω 2V ,i xi

2



EN = %M B(AN ) +

13+

49 B(AN )

⎡

⎣⎢

⎤

⎦⎥

B(AN ) =AN + AN

2 −3227

2

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

23

AN =C (N +32),C =

2 ω%M

,

N =2n+ l =0,1,….

Energy Eigenvalues in the Spin Symmetry and Spherical Symmetry Limit



EN ≈ %M (1+

C (N +32)

2+L )

For the mass large compared to the potential, the eigenenergies go approximately like

that is linearly with N like the non-relativistic spectrum. For the mass small the spectrum goes like

or approximately like N to the 2/3 power. Therefore the harmonic oscillator in the relativistic limit is not harmonic!

Energy Eigenvalues in the Spin and Spherical Symmetry Limit



EN = %M B(AN ) +

13+

49 B(AN )

⎡

⎣⎢

⎤

⎦⎥

=EN − M



Spherical Relativistic Harmonic OscillatorU(3)

In these symmetry limits the energy depends only on N, as in the non-relativistic harmonic oscillator. In the non-

relativistic case this is because the Hamiltonian has an U(3) symmetry.

Although the energy spectrum dependence on N is different than in the non-relativistic case, the Dirac Hamiltonian for

the spherical harmonic oscillator has been shown to have an U(3) symmetry as well in the symmetry limits.

JNG, PRL 95, 252501 (2005)



Generators of Spin and U(3) Symmetry

rS =

rs 00 Up

rsUp

⎛

⎝⎜

⎞

⎠⎟,

rL =

rl 00 Up

rl Up

⎛

⎝⎜⎜

⎞

⎠⎟⎟

rs =

rσ / 2,

rl =

(rr ×

rp)

h,Up =

rσ ·

rp

p.



Quadrupole Generators

qm =

1hMω

32(M 2ω 2[rr]m

(2) +[ pp]m(2))

Qm =qm 0

0 Up qmUp

⎛

⎝⎜⎜

⎞

⎠⎟⎟

From the examples of the spin and orbital angular mometum the following ansatz seems plausible but, in fact, does not work:

Non-relativistic quadrupole generator:



Quadrupole Generators

Qm =(Qm)11 (Qm)12

rσ ·

rp

rσ ·

rp(Qm)21

rσ ·

rp(Qm)22

rσ ·

rp

⎛

⎝⎜⎜

⎞

⎠⎟⎟,



Conditions for Generators to Commute with the Hamiltonian

[Qm , H ]=0



Conditions for Generators to Commute with the Hamiltonian

[Qm , H ]=0implies that

(Qm )12 =(Qm)21,

2[(Qm)11,V] + [(Qm)12 , p2 ] =0,

2[(Qm)12 ,V] + [(Qm)22 , p2 ] =0,

(Qm)11 =(Qm)12 2(V + M ) + (Qm)22 p2 .



A solution is:

Qm =λ2

Mω 2 (Mω 2 r2 + 2M )[rr]m(2) +[ pp]m

(2) Mω 2 [rr]m(2) rσ ·

rp

rσ ·

rpMω 2 [rr]m

(2) [ pp]m(2)

⎛

⎝⎜⎜

⎞

⎠⎟⎟



U(3) Commutation Relations

[N̂ ,rL]=[ N̂,Qm] =0,

[rL,

rL](t) =− 2

rL δt,1,

[rL,Q](t) =− 6 Qδt,2 ,

[Q,Q](t) =3 10rL δt,1.



A solution is:

Qm =λ2

Mω 2 (Mω 2 r2 + 2M )[rr]m(2) +[ pp]m

(2) Mω 2 [rr]m(2) rσ ·

rp

rσ ·

rpMω 2 [rr]m

(2) [ pp]m(2)

⎛

⎝⎜⎜

⎞

⎠⎟⎟

where the norm is determined by the by the commutation relations

λ2 =

3

Mω 2h2 (H + M )

λ0 =

1

2 h (H + M )Mω 2



Number of Quanta Operator

N̂ =λ0

Mω 2 (Mω 2 r2 + 2M )r2 + p2 Mω 2 r2rσ ·

rp

rσ ·

rpMω 2 r2 p2

⎛

⎝⎜⎜

⎞

⎠⎟⎟−32

This can be simplified to



N̂ =

H + M (H −M )

2h Mω 2−32

N∞ =

H −M (H + M )

2h Mω 2−32.



%rS =

UprsUp 0

0rs

⎛

⎝⎜

⎞

⎠⎟, %

rL =

Up

rl Up 0

0rl

⎛

⎝⎜⎜

⎞

⎠⎟⎟,

Pseudo U(3) Generators

%Qm =3

Mω 2h2 ( %H −M )

[ pp]m(2) r

σ ·rpMω 2 [rr]m

(2)

Mω 2 [rr]m(2) rσ ·

rp Mω 2 (Mω 2 r2 −2M )[rr]m

(2) +[ pp]m(2)

⎛

⎝⎜⎜

⎞

⎠⎟⎟



Bound States in the Pseudospin Limit

In the pseudospin limit there are no bound valence Dirac states. The bound states are Dirac hole states. Therefore, even though nuclei are near this limit, it is impossible to do perturbation theory around the exact limit, since in the exact pseudospin symmetry limit there are no bound states. In fact, that is what makes this limit fascinating. This also is a motivation to solve analytically the harmonic oscillator with general vector and scalar potentials so that the system can be studied analytically as the pseudospin limit is approached.



Summary The relativistic harmonic oscillator has a spin and U(3)

symmetry when the scalar and vector potentials are equal. This limit is relevant for hadrons and anti-nucleons in a nuclear environment and perturbation theory is possible.

The relativistic harmonic oscillator has a pseudospin and a pseudo U(3) symmetry when the scalar and vector potentials are equal in magnitude and opposite in sign. This limit is relevant for nuclei, but, in the exact limit, there are no bound Dirac valence states so perturbation is not an option.

Therefore we are attempting to solve analytically the relativistic harmonic oscillator with arbitrary vector and scalar potentials.



Relativistic Harmonic Oscillator with no symmetries: VS (

rr ) ≠±VV (

rr )

The pseudospin limit is approximately valid for nuclei. However in this limit there are no bound Dirac valence states, only bound Dirac hole states. Therefore we would like to solve analytically the Dirac Hamiltonian for general scalar and vector harmonic oscillator potentials to obtain eigenfunctions with a realistic spectrum relevant for nuclei.

Furthermore, this is an interesting problem because there exists a symmetry limit for which perturbation theory is not valid even though

as is the case for nuclei. VS (

rr ) ≈−VV (

rr )



κ =−( j +1

2)

Spherical Relativistic Harmonic Oscillator

is the conserved quantum number for the spherical harmonicharmonic oscillator

for aligned

for unaligned

κ

j =l +

12

κ =( j +1

2)

j =l −

12

The Dirac Hamiltonian leads to two coupled linear equations for the upper and lower components. We do the usual transformation to a second order differential equation for the upper component

gκ (r) =r|κ |−1Gκ (x),x=λr2



Differential Equation

′′G (x) + α (x) ′G (x) + β (x)G(x) = 0

α (x) =2 |κ | +1

2x−

a1

A(x),β (x) =

1

4λ(β−1

x+ β0 + β1x)

A(x) = a0 + a1x,a0 = Eκ + M ,a1 =M

2λ(ωS

2 − ωV2 )x

β−1 = a0b0 ,β0 = a0b1 + a1b0 ,β1 = a1b1

b0 = Eκ − M ,b1 = −M

2λ(ωS

2 + ωV2 )x



Behavior at Long Distances

For short distances the eigenfunctions behave like Gaussians.

However for long distance , and thus

the Airy function ( ).

We are continuing this project.

′′G (x) + b1xG(x) = 0

G(x)⇒ Ai((−b1)13 x)

b1 < 0



Summary

The relativistic harmonic oscillator has a spin symmetry when the scalar and vector potentials are equal. This limit is relevant for hadrons.

The relativistic harmonic oscillator has a pseudospin symmetry when the scalar and vector potentials are equal in magnitude and opposite in sign. This limit is relevant for nuclei, but, in the exact limit, there are no bound Dirac valence states.

Therefore we are attempting to solve analytically the relativistic harmonic oscillator with arbitrary vector and scalar potentials.



Occurs in nuclei

J.N. Ginocchio, Phys. Rev. Lett. 78 (1997) 436

JNG, Phys. Rep 414, 165 (2005)

Slide 5Slide 4

Pseudospin Symmetry Limit

VS (rr ) =−VV (

rr )



Occurs in hadrons

P.R. Page, T. Goldman, J.N. Ginocchio, Phys. Rev. Lett. 86 (2001) 204

Slide 4

Spin Symmetry Limit

VS (rr ) =VV (

rr )

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